where $\mathbf{SINR_l(\mathbf{p})}=G_{ll}p_l/(\sum_{j\ne l} G_{lj}p_j+n_l)$ for all $l$, and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{L})^\top \ge 0$ is a given weight vector to reflect priority among users (larger weight means higher priority). A total power budget is given by $\bar{P}$.